The delicate point,
as observed by Beekie-Buckmaster-Vicol [1]},
is that the magnetic helicity is conserved under much milder regularity conditions.
As a matter of fact,
the magnetic helicity is commonly expected in the plasma physics to be conserved in the infinite conductivity limit,
known as Taylor's conjecture ([2]}, [3]}).
This conjecture was proved by Faraco and Lindberg [4]}
under the weak ideal limits,
namely, the weak limits of Leray solutions to MHD (REF )
when \({\alpha }_i=1, i=1,2\) .
For general weak solutions,
Beekie-Buckmaster-Vicol [1]} first constructed weak solutions
in \(C_tL^2_x\)  violating the magnetic helicity conservation.
Unlike in [6]}, the proof of [1]} relies on the intermittent convex integration scheme,
in which the intermittent shear flows are constructed to respect the geometry of MHD.
See also Dai [8]} for the Hall MHD system
where the Hall nonlinearity takes the dominant effect,
and Feireisl-Li [9]} for the ill-posedness of the MHD system
where the fluid is compressible, inviscid and magnetically resistive.
